In this article, we apply the coupled-mode theory to vertically-coupled micro-disk resonators presenting an asymmetric distribution of refractive index and a multilayer separation region between the two waveguide cores, resulting in an effective propagation constant phase-mismatch in the coupling region. We introduce a criterion which, given the coupler overall permittivity distribution, clarifies how to best choose the individual decomposition index profiles among the various possible solutions. Following our recent experimental demonstration we subsequently exploit the derived decomposition to evaluate the theoretical transmission characteristics of an AlGaAs/AlOx-based structure as function of wavelength and as function of the position of the resonator relative to the access waveguide.We show that the resonant dips of the intensity transmission, spaced by the cavity FSR, are modulated by an envelop which governs the coupling regime of the resonator-waveguide system.